A GRG Formulation: Circle and Area from a Binary Radial Framework
Generative Radial Geometry describes circular measures through a finite binary radial framework. Its working quantities are a radius, a full Turn, a binary resolution, and local chord–tangent constructions.
About the Current GRG Formulation
This page presents the documented binary radial formulation, with geometric definitions and an explicit verification of its measurement kernels. Earlier approaches, including Original Projection and previous Turns formulations, remain part of the research history.
The geometric construction, kernel accuracy, and computational efficiency are separate subjects. The analysis below establishes accuracy for regular symmetric sectors. It does not establish a performance advantage or prescribe an implementation.
The weighted area proposal is documented in A Symmetric GRG Kernel for Area. The geometric ratio notation and proofs below are an analytical clarification of the documented triangles and kernels.
Binary Turns and Resolution
A full rotation is one Turn. Resolve it into equal sectors. The nondegenerate chord–tangent formulas on this page require at least four sectors.
N=2^b,\qquad b\ge2,\qquad R>0.
$$
t_b=\frac1N,\qquad \text{half-sector width}=\frac1{2N}.
$$
One Symmetric Radial Sector
Let a be the distance from the centre to the internal chord, h half that chord, and e half the external tangent side. The tangent is perpendicular to the central radius. The two boundary rays intersect both the circle and this tangent.
a^2+h^2=R^2,\qquad e=R\frac ha.
$$
Define dimensionless ratios from these constructed lengths:
c_b=\frac aR,\qquad s_b=\frac hR,\qquad q_b=\frac ha.
$$
c_b^2+s_b^2=1,\qquad q_b=\frac{s_b}{c_b},\qquad c_b=\frac1{\sqrt{1+q_b^2}}.
$$
These symbols denote geometric ratios. They do not specify a trigonometric software routine.
Circumference Kernels
L_i=2h,\qquad L_t=2e.
$$
The arithmetic balance and the classical Snell–Huygens weighted balance give:
\begin{aligned}C_m&=\frac N2(L_i+L_t)=NR(s_b+q_b),\\C_{SH}&=\frac N3(2L_i+L_t)=\frac{2NR}{3}(2s_b+q_b).\end{aligned}
$$
The weighting is a known local arc approximation. This formulation uses it as a kernel within the binary radial framework; the weighting itself is not claimed as an original GRG result.
Area Elements and Kernels
The internal triangle has base 2h and height a. The external triangle has base 2e and height R.
\begin{aligned}I_b&=ah=R^2c_bs_b=R^2\frac{q_b}{1+q_b^2},\\E_b&=Re=R^2q_b.\end{aligned}
$$
A_{\mathrm{in}}=NI_b,\qquad A_{\mathrm{out}}=NE_b.
$$
The arithmetic area estimate and the proposed internal–external weighting are:
\begin{aligned}A_m&=\frac N2(I_b+E_b),\\A_w&=\frac N3(I_b+2E_b).\end{aligned}
$$
A_w=\frac{NR^2}{3}\left(\frac{q_b}{1+q_b^2}+2q_b\right).
$$
Geometric Refinement Bound
The following identity is used to verify regular equal-sector bisection. It is a geometric consequence, not an attribution of a particular numerical algorithm to GRG.
At four sectors the half-sector triangle has equal legs, so q₂ = 1. Bisecting its direction gives the following ratio, obtained by normalising the boundary direction and adding it to the central unit direction:
q_{b+1}=\frac{q_b}{1+\sqrt{1+q_b^2}}.
$$
The denominator is at least two. Induction therefore gives:
0\lt q_b\le 2^{2-b}=\frac4N.
$$
This proves the scale bound needed for the global error estimates. It uses the regular bisection geometry; it does not require a precomputed full-Turn length or area.
Kernel Accuracy: Analytical Verification
Define the local integral below solely as an analytical reference for the exact sector measures. It is not an input to the measurement kernels.
F(q)=\int_0^q\frac{dt}{1+t^2}.
$$
Parameterising the circular boundary by its slope gives exact sector arc length 2RF(q) and exact sector area R²F(q). Thus the fixed full-Turn reference measures satisfy:
C_T(R)=2NRF(q_b),\qquad A_T(R)=NR^2F(q_b).
$$
F(q)=q-\frac{q^3}{3}+\frac{q^5}{5}+O(q^7).
$$
Substitution into the circumference kernels yields:
\begin{aligned}C_m-C_T&=NR\left(\frac{q_b^3}{6}-\frac{q_b^5}{40}+O(q_b^7)\right),\\C_{SH}-C_T&=NR\left(\frac{q_b^5}{10}+O(q_b^7)\right).\end{aligned}
$$
For area:
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$
The area weighting cancels the cubic local residual. Combining these expansions with the geometric scale bound proves, for fixed radius and exact arithmetic:
\begin{aligned}C_m-C_T&=O(N^{-2}),& C_{SH}-C_T&=O(N^{-4}),\\A_m-A_T&=O(N^{-2}),& A_w-A_T&=O(N^{-4}).\end{aligned}
$$
These are asymptotic error statements, not claims of exact equality at finite resolution or guarantees about floating-point performance.
Area Bounds
The geometric inner and outer triangles bound each exact sector. The two balanced area estimates provide a tighter bracket as well:
A_{\mathrm{in}}\le A_m\le A_T\le A_w\le A_{\mathrm{out}}.
$$
For a direct verification, write the normalised kernels as M(q) and W(q). Both differences vanish at zero, and:
\begin{aligned}\frac{d}{dq}\bigl(F(q)-M(q)\bigr)&=\frac{q^2(1-q^2)}{2(1+q^2)^2},\\\frac{d}{dq}\bigl(W(q)-F(q)\bigr)&=\frac{2q^4}{3(1+q^2)^2}.\end{aligned}
$$
Both derivatives are nonnegative for 0 ≤ q ≤ 1, which covers the sectors considered here. Consequently the bracket follows for every b ≥ 2, rather than only for tested resolutions.
Refinement and Quarter-Step Sampling
N_{b+1}=2N_b,\qquad t_{b+1}=\frac{t_b}{2},\qquad \Delta u=\frac{t_b}{4}=\frac1{4N}.
$$
The related area article specifies quarter-step sampling within a module. The verification above concerns the complete sector triangles. An implementation using internal samples must specify how those samples evaluate the same elements; the sampling parameter alone does not define an additional area kernel.
Computational Efficiency and Research Scope
Binary resolution is compatible with binary angular indices and power-of-two scaling. A performance advantage for the complete calculation must nevertheless be measured.
A reproducible comparison must use equal accuracy targets, include generation or retrieval of the local measures, account for precomputation and memory, and specify arithmetic precision, rounding, hardware, and execution time. Stability in fixed-point arithmetic requires a separate error analysis.
The results established here concern the documented regular chord–tangent geometry and its kernels. Historical originality and implementation efficiency are separate investigations.